A Fractional‐Order System with Coexisting Chaotic Attractors and Control Chaos via a Single State Variable Linear Controller

Zhou, Ping; Ke, Meihua; Pan, Zhu

doi:10.1155/2018/4192824

Cited by 4 publications

(1 citation statement)

References 20 publications

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“…Similarly to integer order chaotic systems, the control and synchronization of fractional-order chaotic systems has become an active research field [8][9][10][11][12][13][14][15]. It is not difficult to see that in papers [8][9][10][11][12][13][14][15] the authors have used all state variables to design controllers. However, in the real situation it is well known that only part of the variables can be used in many nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

The Robust Control and Synchronization of a Class of Fractional‐Order Chaotic Systems with External Disturbances via a Single Output

Luo

2018

Complexity

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This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.

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Section: Introductionmentioning

confidence: 99%

The Robust Control and Synchronization of a Class of Fractional‐Order Chaotic Systems with External Disturbances via a Single Output

Luo

2018

Complexity

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A fractional model for predator-prey with omnivore

Bonyah

Atangana

Elsadany

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard’s method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems.

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Breaking of integrability and conservation leading to Hamiltonian chaotic system and its energy-based coexistence analysis

Gou

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, a four-dimensional conservative system of Euler equations producing the periodic orbit is constructed and studied. The reason that a conservative system often produces periodic orbit has rarely been studied. By analyzing the Hamiltonian and Casimir functions, three invariants of the conservative system are found. The complete integrability is proved to be the mechanism that the system generates the periodic orbits. The mechanism route from periodic orbit to conservative chaos is found by breaking the conservation of Casimir energy and the integrability through which a chaotic Hamiltonian system is built. The observed chaos is not excited by saddle or center equilibria, so the system has hidden dynamics. It is found that the upgrade in the Hamiltonian energy level violates the order of dynamical behavior and transitions from a low or regular state to a high or an irregular state. From the energy bifurcation associated with different energy levels, rich coexisting orbits are discovered, i.e., the coexistence of chaotic orbits, quasi-periodic orbits, and chaotic quasi-periodic orbits. The coincidence between the two-dimensional diagram of maximum Lyapunov exponents and the bifurcation diagram of Hamiltonian energy is observed. Finally, field programmable gate array implementation, a challenging task for the chaotic Hamiltonian conservative system, is designed to be a Hamiltonian pseudo-random number generator.

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A Fractional‐Order System with Coexisting Chaotic Attractors and Control Chaos via a Single State Variable Linear Controller

Cited by 4 publications

References 20 publications

The Robust Control and Synchronization of a Class of Fractional‐Order Chaotic Systems with External Disturbances via a Single Output

The Robust Control and Synchronization of a Class of Fractional‐Order Chaotic Systems with External Disturbances via a Single Output

A fractional model for predator-prey with omnivore

Breaking of integrability and conservation leading to Hamiltonian chaotic system and its energy-based coexistence analysis

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